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It is well known that, for radiating structures supporting scalar spherical modes of index nN, the maximum obtainable directivity is limited to (N+1)2; the pattern functions for the corresponding arrangement of excitation coefficients are found in closed form. Alternate measures of beam concentration are formulated and optimum pattern functions are computed in closed form. These alternate measures, which are here called beam spread, depend on the entire radiation pattern (weighted by an appropriately chosen weight factor) and are reciprocal to the measures of directivity proposed by Synge (1966). The radiation patterns corresponding to optimizing the classical directivity and beam spread, with respect to a weight function (1 - cos θ), are compared, and it is found that optimizing beam spread yields superior side-lobe structure at the expense of a slight loss in directivity. The extension of these techniques to other weight functions is also explored.